In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism f : X → Y which is rightcancellative in the sense that, for all morphisms g_{1}, g_{2} : Y → Z,
Epimorphisms are analogues of surjective functions, but they are not exactly the same. The dual of an epimorphism is a monomorphism (i.e. an epimorphism in a category C is a monomorphism in the dual category C^{op}).
Many authors in abstract algebra and universal algebra define an epimorphism simply as an onto or surjective homomorphism. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. In this article, the term "epimorphism" will be used in the sense of category theory given above. For more on this, see the section on Terminology below.
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Examples
Every morphism in a concrete category whose underlying function is surjective is an epimorphism. In many concrete categories of interest the converse is also true. For example, in the following categories, the epimorphisms are exactly those morphisms which are surjective on the underlying sets:
 Set, sets and functions. To prove that every epimorphism f: X → Y in Set is surjective, we compose it with both the characteristic function g_{1}: Y → {0,1} of the image f(X) and the map g_{2}: Y → {0,1} that is constant 1.
 Rel, sets with binary relations and relation preserving functions. Here we can use the same proof as for Set, equipping {0,1} with the full relation {0,1}×{0,1}.
 Pos, partially ordered sets and monotone functions. If f : (X,≤) → (Y,≤) is not surjective, pick y_{0} in Y \ f(X) and let g_{1} : Y → {0,1} be the characteristic function of {y  y_{0} ≤ y} and g_{2} : Y → {0,1} the characteristic function of {y  y_{0} < y}. These maps are monotone if {0,1} is given the standard ordering 0 < 1.
 Grp, groups and group homomorphisms. The result that every epimorphism in Grp is surjective is due to Otto Schreier (he actually proved more, showing that every subgroup is an equalizer using the free product with one amalgamated subgroup); an elementary proof can be found in (Linderholm 1970).
 FinGrp, finite groups and group homomorphisms. Also due to Schreier; the proof given in (Linderholm 1970) establishes this case as well.
 Ab, abelian groups and group homomorphisms.
 KVect, vector spaces over a field K and Klinear transformations.
 ModR, right modules over a ring R and module homomorphisms. This generalizes the two previous examples; to prove that every epimorphism f: X → Y in ModR is surjective, we compose it with both the canonical quotient map g _{1}: Y → Y/f(X) and the zero map g_{2}: Y → Y/f(X).
 Top, topological spaces and continuous functions. To prove that every epimorphism in Top is surjective, we proceed exactly as in Set, giving {0,1} the indiscrete topology which ensures that all considered maps are continuous.
 HComp, compact Hausdorff spaces and continuous functions. If f: X → Y is not surjective, let y in YfX. Since fX is closed, by Urysohn's Lemma there is a continuous function g_{1}:Y → [0,1] such that g_{1} is 0 on fX and 1 on y. We compose f with both g_{1} and the zero function g_{2}: Y → [0,1].
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